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Vector Algebra — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsVector Algebra1 Mark
MCQ
Choose the correct answer from the given four options.
If $|\vec{{\text{a}}}|=10,|\vec{{\text{b}}}|=2$ and $\vec{{\text{a}}}\cdot\vec{{\text{b}}}=12,$ then value of $|\vec{{\text{a}}}\times\vec{\text{b}}|$ is :
  • A
    $5.$
  • B
    $10.$
  • C
    $14.$
  • ✓
    $16.$
✓

Answer

Correct option: D.
$16.$
Here, $|\vec{{\text{a}}}|=10,|\vec{{\text{b}}}|=2$ and $\vec{{\text{a}}}\cdot\vec{\text{b}}=12\ [$given$]$
$\therefore\vec{{\text{a}}}\cdot\vec{\text{b}}=|\vec{{\text{a}}}||\vec{{\text{b}}}|\cos\theta$
$12=10\times2\cos\theta$
$\Rightarrow\cos\theta=\frac{12}{20}=\frac{3}{5}$
$\Rightarrow\sin\theta=\sqrt{1-\cos\theta}$
$=\sqrt{1-\frac{9}{25}}$
$\sin\theta=\pm\frac{4}{5}$
$\therefore|\vec{{\text{a}}}\times\vec{{\text{b}}}|=|\vec{{\text{a}}}\|\vec{{\text{b}}}\|\sin\theta|$
$=10\times2\times\frac{4}{5}$
$=16$

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